How to make "Weight Paint" affect all vertices, even those with zero weight?
A smooth deformation of a bunch of vertices is always due to a balancing between 2 or more different weight assignements: if a vertex is weighted to a vertex group only, it will move with its relative bone rigidly and exactly, even if the weight is 0.00000001. Instead, if it has some weight relatively to two (or more) vertex groups, the movement is calculated making a comparision between the two values, with an automatic normalization; normalisation means that if one value is 0.001 and the other is 0.009, they are treated as 0.1 and 0.9 and their sum will be treated as 1.0.

So it's importany to analyze all weights, in your case not only head bone but neck bone also. If you select a weighted vertex in edit mode, you will find in the "N" properties panel a "vertex weights" tab in which you can inspect all weights and have a better understanding of the behaviour.

To solve your problem you will have to give some weight to the neck vertex group, so that vertices near the neck will be pushed to two different actions: the head bone will try to move them while the neck bone will try to make them stay in place.

Image is 2.79 but 2.81 is the same.